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# Power series method and approximate linear differential equations of second order

DOI: 10.1186/1687-1847-2013-76

Accepted: 4 March 2013

Published: 26 March 2013

## Abstract

In this paper, we will establish a theory for the power series method that can be applied to various types of linear differential equations of second order to prove the Hyers-Ulam stability.

MSC:34A05, 39B82, 26D10, 34A40.

### Keywords

power series method approximate linear differential equation simple harmonic oscillator equation Hyers-Ulam stability approximation

## 1 Introduction

Let X be a normed space over a scalar field $\mathbb{K}$, and let $I\subset \mathbb{R}$ be an open interval, where $\mathbb{K}$ denotes either or . Assume that ${a}_{0},{a}_{1},\dots ,{a}_{n}:I\to \mathbb{K}$ and $g:I\to X$ are given continuous functions. If for every n times continuously differentiable function $y:I\to X$ satisfying the inequality
$\parallel {a}_{n}\left(x\right){y}^{\left(n\right)}\left(x\right)+{a}_{n-1}\left(x\right){y}^{\left(n-1\right)}\left(x\right)+\cdots +{a}_{1}\left(x\right){y}^{\prime }\left(x\right)+{a}_{0}\left(x\right)y\left(x\right)+g\left(x\right)\parallel \le \epsilon$
for all $x\in I$ and for a given $\epsilon >0$, there exists an n times continuously differentiable solution ${y}_{0}:I\to X$ of the differential equation
${a}_{n}\left(x\right){y}^{\left(n\right)}\left(x\right)+{a}_{n-1}\left(x\right){y}^{\left(n-1\right)}\left(x\right)+\cdots +{a}_{1}\left(x\right){y}^{\prime }\left(x\right)+{a}_{0}\left(x\right)y\left(x\right)+g\left(x\right)=0$

such that $\parallel y\left(x\right)-{y}_{0}\left(x\right)\parallel \le K\left(\epsilon \right)$ for any $x\in I$, where $K\left(\epsilon \right)$ is an expression of ε with ${lim}_{\epsilon \to 0}K\left(\epsilon \right)=0$, then we say that the above differential equation has the Hyers-Ulam stability. For more detailed definitions of the Hyers-Ulam stability, we refer the reader to [18].

Obłoza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see [9, 10]). Thereafter, Alsina and Ger [11] proved the Hyers-Ulam stability of the differential equation ${y}^{\prime }\left(x\right)=y\left(x\right)$. It was further proved by Takahasi et al. that the Hyers-Ulam stability holds for the Banach space valued differential equation ${y}^{\prime }\left(x\right)=\lambda y\left(x\right)$ (see [12] and also [1315]).

Moreover, Miura et al. [16] investigated the Hyers-Ulam stability of an n th-order linear differential equation. The first author also proved the Hyers-Ulam stability of various linear differential equations of first order (ref. [1725]).

Recently, the first author applied the power series method to studying the Hyers-Ulam stability of several types of linear differential equations of second order (see [2634]). However, it was inconvenient that he had to alter and apply the power series method with respect to each differential equation in order to study the Hyers-Ulam stability. Thus, it is inevitable to develop a power series method that can be comprehensively applied to different types of differential equations.

In Sections 2 and 3 of this paper, we establish a theory for the power series method that can be applied to various types of linear differential equations of second order to prove the Hyers-Ulam stability.

Throughout this paper, we assume that the linear differential equation of second order of the form
$p\left(x\right){y}^{″}\left(x\right)+q\left(x\right){y}^{\prime }\left(x\right)+r\left(x\right)y\left(x\right)=0,$
(1)
for which $x=0$ is an ordinary point, has the general solution ${y}_{h}:\left(-{\rho }_{0},{\rho }_{0}\right)\to \mathbb{C}$, where ${\rho }_{0}$ is a constant with $0<{\rho }_{0}\le \mathrm{\infty }$ and the coefficients $p,q,r:\left(-{\rho }_{0},{\rho }_{0}\right)\to \mathbb{C}$ are analytic at 0 and have power series expansions
$p\left(x\right)=\sum _{m=0}^{\mathrm{\infty }}{p}_{m}{x}^{m},\phantom{\rule{2em}{0ex}}q\left(x\right)=\sum _{m=0}^{\mathrm{\infty }}{q}_{m}{x}^{m}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}r\left(x\right)=\sum _{m=0}^{\mathrm{\infty }}{r}_{m}{x}^{m}$

for all $x\in \left(-{\rho }_{0},{\rho }_{0}\right)$. Since $x=0$ is an ordinary point of (1), we remark that ${p}_{0}\ne 0$.

## 2 Inhomogeneous differential equation

In the following theorem, we solve the linear inhomogeneous differential equation of second order of the form
$p\left(x\right){y}^{″}\left(x\right)+q\left(x\right){y}^{\prime }\left(x\right)+r\left(x\right)y\left(x\right)=\sum _{m=0}^{\mathrm{\infty }}{a}_{m}{x}^{m}$
(2)

under the assumption that $x=0$ is an ordinary point of the associated homogeneous linear differential equation (1).

Theorem 2.1 Assume that the radius of convergence of power series ${\sum }_{m=0}^{\mathrm{\infty }}{a}_{m}{x}^{m}$ is ${\rho }_{1}>0$ and that there exists a sequence $\left\{{c}_{m}\right\}$ satisfying the recurrence relation
$\sum _{k=0}^{m}\left[\left(k+2\right)\left(k+1\right){c}_{k+2}{p}_{m-k}+\left(k+1\right){c}_{k+1}{q}_{m-k}+{c}_{k}{r}_{m-k}\right]={a}_{m}$
(3)
for any $m\in {\mathbb{N}}_{0}$. Let ${\rho }_{2}$ be the radius of convergence of power series ${\sum }_{m=0}^{\mathrm{\infty }}{c}_{m}{x}^{m}$ and let ${\rho }_{3}=min\left\{{\rho }_{0},{\rho }_{1},{\rho }_{2}\right\}$, where $\left(-{\rho }_{0},{\rho }_{0}\right)$ is the domain of the general solution to (1). Then every solution $y:\left(-{\rho }_{3},{\rho }_{3}\right)\to \mathbb{C}$ of the linear inhomogeneous differential equation (2) can be expressed by
$y\left(x\right)={y}_{h}\left(x\right)+\sum _{m=0}^{\mathrm{\infty }}{c}_{m}{x}^{m}$

for all $x\in \left(-{\rho }_{3},{\rho }_{3}\right)$, where ${y}_{h}\left(x\right)$ is a solution of the linear homogeneous differential equation (1).

Proof Since $x=0$ is an ordinary point, we can substitute ${\sum }_{m=0}^{\mathrm{\infty }}{c}_{m}{x}^{m}$ for $y\left(x\right)$ in (2) and use the formal multiplication of power series and consider (3) to get
$\begin{array}{c}p\left(x\right){y}^{″}\left(x\right)+q\left(x\right){y}^{\prime }\left(x\right)+r\left(x\right)y\left(x\right)\hfill \\ \phantom{\rule{1em}{0ex}}=\sum _{m=0}^{\mathrm{\infty }}\sum _{k=0}^{m}{p}_{m-k}\left(k+2\right)\left(k+1\right){c}_{k+2}{x}^{m}+\sum _{m=0}^{\mathrm{\infty }}\sum _{k=0}^{m}{q}_{m-k}\left(k+1\right){c}_{k+1}{x}^{m}\hfill \\ \phantom{\rule{2em}{0ex}}+\sum _{m=0}^{\mathrm{\infty }}\sum _{k=0}^{m}{r}_{m-k}{c}_{k}{x}^{m}\hfill \\ \phantom{\rule{1em}{0ex}}=\sum _{m=0}^{\mathrm{\infty }}\sum _{k=0}^{m}\left[\left(k+2\right)\left(k+1\right){c}_{k+2}{p}_{m-k}+\left(k+1\right){c}_{k+1}{q}_{m-k}+{c}_{k}{r}_{m-k}\right]{x}^{m}\hfill \\ \phantom{\rule{1em}{0ex}}=\sum _{m=0}^{\mathrm{\infty }}{a}_{m}{x}^{m}\hfill \end{array}$
for all $x\in \left(-{\rho }_{3},{\rho }_{3}\right)$. That is, ${\sum }_{m=0}^{\mathrm{\infty }}{c}_{m}{x}^{m}$ is a particular solution of the linear inhomogeneous differential equation (2), and hence every solution $y:\left(-{\rho }_{3},{\rho }_{3}\right)\to \mathbb{C}$ of (2) can be expressed by
$y\left(x\right)={y}_{h}\left(x\right)+\sum _{m=0}^{\mathrm{\infty }}{c}_{m}{x}^{m},$

where ${y}_{h}\left(x\right)$ is a solution of the linear homogeneous differential equation (1). □

For the most common case in applications, the coefficient functions $p\left(x\right)$, $q\left(x\right)$, and $r\left(x\right)$ of the linear differential equation (1) are simple polynomials. In such a case, we have the following corollary.

Corollary 2.2 Let $p\left(x\right)$, $q\left(x\right)$, and $r\left(x\right)$ be polynomials of degree at most $d\ge 0$. In particular, let ${d}_{0}$ be the degree of $p\left(x\right)$. Assume that the radius of convergence of power series ${\sum }_{m=0}^{\mathrm{\infty }}{a}_{m}{x}^{m}$ is ${\rho }_{1}>0$ and that there exists a sequence $\left\{{c}_{m}\right\}$ satisfying the recurrence formula
$\sum _{k={m}_{0}}^{m}\left[\left(k+2\right)\left(k+1\right){c}_{k+2}{p}_{m-k}+\left(k+1\right){c}_{k+1}{q}_{m-k}+{c}_{k}{r}_{m-k}\right]={a}_{m}$
(4)
for any $m\in {\mathbb{N}}_{0}$, where ${m}_{0}=max\left\{0,m-d\right\}$. If the sequence $\left\{{c}_{m}\right\}$ satisfies the following conditions:
1. (i)

${lim}_{m\to \mathrm{\infty }}{c}_{m-1}/m{c}_{m}=0$,

2. (ii)

there exists a complex number L such that ${lim}_{m\to \mathrm{\infty }}{c}_{m}/{c}_{m-1}=L$ and ${p}_{{d}_{0}}+L{p}_{{d}_{0}-1}+\cdots +{L}^{{d}_{0}-1}{p}_{1}+{L}^{{d}_{0}}{p}_{0}\ne 0$,

then every solution $y:\left(-{\rho }_{3},{\rho }_{3}\right)\to \mathbb{C}$ of the linear inhomogeneous differential equation (2) can be expressed by
$y\left(x\right)={y}_{h}\left(x\right)+\sum _{m=0}^{\mathrm{\infty }}{c}_{m}{x}^{m}$

for all $x\in \left(-{\rho }_{3},{\rho }_{3}\right)$, where ${\rho }_{3}=min\left\{{\rho }_{0},{\rho }_{1}\right\}$ and ${y}_{h}\left(x\right)$ is a solution of the linear homogeneous differential equation (1).

Proof Let m be any sufficiently large integer. Since ${p}_{d+1}={p}_{d+2}=\cdots =0$, ${q}_{d+1}={q}_{d+2}=\cdots =0$ and ${r}_{d+1}={r}_{d+2}=\cdots =0$, if we substitute $m-d+k$ for k in (4), then we have
$\begin{array}{rcl}{a}_{m}& =& \sum _{k=0}^{d}\left[\left(m-d+k+2\right)\left(m-d+k+1\right){c}_{m-d+k+2}{p}_{d-k}\\ +\left(m-d+k+1\right){c}_{m-d+k+1}{q}_{d-k}+{c}_{m-d+k}{r}_{d-k}\right].\end{array}$
By (i) and (ii), we have
$\begin{array}{c}\underset{m\to \mathrm{\infty }}{lim sup}{|{a}_{m}|}^{1/m}\hfill \\ \phantom{\rule{1em}{0ex}}=\underset{m\to \mathrm{\infty }}{lim sup}|\sum _{k=0}^{d}\left(m-d+k+2\right)\left(m-d+k+1\right){c}_{m-d+k+2}\hfill \\ \phantom{\rule{2em}{0ex}}×\left({p}_{d-k}+\frac{{q}_{d-k}}{\left(m-d+k+2\right)}\frac{{c}_{m-d+k+1}}{{c}_{m-d+k+2}}\hfill \\ \phantom{\rule{2em}{0ex}}+{\frac{{r}_{d-k}}{\left(m-d+k+2\right)\left(m-d+k+1\right)}\frac{{c}_{m-d+k}}{{c}_{m-d+k+1}}\frac{{c}_{m-d+k+1}}{{c}_{m-d+k+2}}\right)|}^{1/m}\hfill \\ \phantom{\rule{1em}{0ex}}=\underset{m\to \mathrm{\infty }}{lim sup}|\sum _{k=0}^{d}\left(m-d+k+2\right)\left(m-d+k+1\right){c}_{m-d+k+2}{p}_{d-k}{|}^{1/m}\hfill \\ \phantom{\rule{1em}{0ex}}=\underset{m\to \mathrm{\infty }}{lim sup}|\sum _{k=d-{d}_{0}}^{d}\left(m-d+k+2\right)\left(m-d+k+1\right){c}_{m-d+k+2}{p}_{d-k}{|}^{1/m}\hfill \\ \phantom{\rule{1em}{0ex}}=\underset{m\to \mathrm{\infty }}{lim sup}|\left(m-{d}_{0}+2\right)\left(m-{d}_{0}+1\right){c}_{m-{d}_{0}+2}\left({p}_{{d}_{0}}+L{p}_{{d}_{0}-1}+\cdots +{L}^{{d}_{0}}{p}_{0}\right){|}^{1/m}\hfill \\ \phantom{\rule{1em}{0ex}}=\underset{m\to \mathrm{\infty }}{lim sup}|\left({p}_{{d}_{0}}+L{p}_{{d}_{0}-1}+\cdots +{L}^{{d}_{0}}{p}_{0}\right)\left(m-{d}_{0}+2\right)\left(m-{d}_{0}+1\right){|}^{1/m}\hfill \\ \phantom{\rule{2em}{0ex}}×{\left({|{c}_{m-{d}_{0}+2}|}^{1/\left(m-{d}_{0}+2\right)}\right)}^{\left(m-{d}_{0}+2\right)/m}\hfill \\ \phantom{\rule{1em}{0ex}}=\underset{m\to \mathrm{\infty }}{lim sup}{|{c}_{m-{d}_{0}+2}|}^{1/\left(m-{d}_{0}+2\right)},\hfill \end{array}$

which implies that the radius of convergence of the power series ${\sum }_{m=0}^{\mathrm{\infty }}{c}_{m}{x}^{m}$ is ${\rho }_{1}$. The rest of this corollary immediately follows from Theorem 2.1. □

In many cases, it occurs that $p\left(x\right)\equiv 1$ in (1). For this case, we obtain the following corollary.

Corollary 2.3 Let ${\rho }_{3}$ be a distance between the origin 0 and the closest one among singular points of $q\left(z\right)$, $r\left(z\right)$, or ${\sum }_{m=0}^{\mathrm{\infty }}{a}_{m}{z}^{m}$ in a complex variable z. If there exists a sequence $\left\{{c}_{m}\right\}$ satisfying the recurrence relation
$\left(m+2\right)\left(m+1\right){c}_{m+2}+\sum _{k=0}^{m}\left[\left(k+1\right){c}_{k+1}{q}_{m-k}+{c}_{k}{r}_{m-k}\right]={a}_{m}$
(5)
for any $m\in {\mathbb{N}}_{0}$, then every solution $y:\left(-{\rho }_{3},{\rho }_{3}\right)\to \mathbb{C}$ of the linear inhomogeneous differential equation
${y}^{″}\left(x\right)+q\left(x\right){y}^{\prime }\left(x\right)+r\left(x\right)y\left(x\right)=\sum _{m=0}^{\mathrm{\infty }}{a}_{m}{x}^{m}$
(6)
can be expressed by
$y\left(x\right)={y}_{h}\left(x\right)+\sum _{m=0}^{\mathrm{\infty }}{c}_{m}{x}^{m}$

for all $x\in \left(-{\rho }_{3},{\rho }_{3}\right)$, where ${y}_{h}\left(x\right)$ is a solution of the linear homogeneous differential equation (1) with $p\left(x\right)\equiv 1$.

Proof If we put ${p}_{0}=1$ and ${p}_{i}=0$ for each $i\in \mathbb{N}$, then the recurrence relation (3) reduces to (5). As we did in the proof of Theorem 2.1, we can show that ${\sum }_{m=0}^{\mathrm{\infty }}{c}_{m}{x}^{m}$ is a particular solution of the linear inhomogeneous differential equation (6).

According to [[35], Theorem 7.4] or [[36], Theorem 5.2.1], there is a particular solution ${y}_{0}\left(x\right)$ of (6) in a form of power series in x whose radius of convergence is at least ${\rho }_{3}$. Moreover, since ${\sum }_{m=0}^{\mathrm{\infty }}{c}_{m}{x}^{m}$ is a solution of (6), it can be expressed as a sum of both ${y}_{0}\left(x\right)$ and a solution of the homogeneous equation (1) with $p\left(x\right)\equiv 1$. Hence, the radius of convergence of ${\sum }_{m=0}^{\mathrm{\infty }}{c}_{m}{x}^{m}$ is at least ${\rho }_{3}$.

Now, every solution $y:\left(-{\rho }_{3},{\rho }_{3}\right)\to \mathbb{C}$ of (6) can be expressed by
$y\left(x\right)={y}_{h}\left(x\right)+\sum _{m=0}^{\mathrm{\infty }}{c}_{m}{x}^{m},$

where ${y}_{h}\left(x\right)$ is a solution of the linear differential equation (1) with $p\left(x\right)\equiv 1$. □

## 3 Approximate differential equation

In this section, let ${\rho }_{1}>0$ be a constant. We denote by $\mathcal{C}$ the set of all functions $y:\left(-{\rho }_{1},{\rho }_{1}\right)\to \mathbb{C}$ with the following properties:
1. (a)

$y\left(x\right)$ is expressible by a power series ${\sum }_{m=0}^{\mathrm{\infty }}{b}_{m}{x}^{m}$ whose radius of convergence is at least ${\rho }_{1}$;

2. (b)
There exists a constant $K\ge 0$ such that ${\sum }_{m=0}^{\mathrm{\infty }}|{a}_{m}{x}^{m}|\le K|{\sum }_{m=0}^{\mathrm{\infty }}{a}_{m}{x}^{m}|$ for any $x\in \left(-{\rho }_{1},{\rho }_{1}\right)$, where
${a}_{m}=\sum _{k=0}^{m}\left[\left(k+2\right)\left(k+1\right){b}_{k+2}{p}_{m-k}+\left(k+1\right){b}_{k+1}{q}_{m-k}+{b}_{k}{r}_{m-k}\right]$

for all $m\in {\mathbb{N}}_{0}$ and ${p}_{0}\ne 0$.

Lemma 3.1 Given a sequence $\left\{{a}_{m}\right\}$, let $\left\{{c}_{m}\right\}$ be a sequence satisfying the recurrence formula (3) for all $m\in {\mathbb{N}}_{0}$. If ${p}_{0}\ne 0$ and $n\ge 2$, then ${c}_{n}$ is a linear combination of ${a}_{0},{a}_{1},\dots ,{a}_{n-2}$, ${c}_{0}$, and ${c}_{1}$.

Proof We apply induction on n. Since ${p}_{0}\ne 0$, if we set $m=0$ in (3), then
${c}_{2}=\frac{1}{2{p}_{0}}{a}_{0}-\frac{{r}_{0}}{2{p}_{0}}{c}_{0}-\frac{{q}_{0}}{2{p}_{0}}{c}_{1},$
i.e., ${c}_{2}$ is a linear combination of ${a}_{0}$, ${c}_{0}$, and ${c}_{1}$. Assume now that n is an integer not less than 2 and ${c}_{i}$ is a linear combination of ${a}_{0},\dots ,{a}_{i-2}$, ${c}_{0}$, ${c}_{1}$ for all $i\in \left\{2,3,\dots ,n\right\}$, namely,
${c}_{i}={\alpha }_{i}^{0}{a}_{0}+{\alpha }_{i}^{1}{a}_{1}+\cdots +{\alpha }_{i}^{i-2}{a}_{i-2}+{\beta }_{i}{c}_{0}+{\gamma }_{i}{c}_{1},$
where ${\alpha }_{i}^{0},\dots ,{\alpha }_{i}^{i-2}$, ${\beta }_{i}$, ${\gamma }_{i}$ are complex numbers. If we replace m in (3) with $n-1$, then
$\begin{array}{rcl}{a}_{n-1}& =& 2{c}_{2}{p}_{n-1}+{c}_{1}{q}_{n-1}+{c}_{0}{r}_{n-1}\\ +6{c}_{3}{p}_{n-2}+2{c}_{2}{q}_{n-2}+{c}_{1}{r}_{n-2}\\ +\cdots \\ +n\left(n-1\right){c}_{n}{p}_{1}+\left(n-1\right){c}_{n-1}{q}_{1}+{c}_{n-2}{r}_{1}\\ +\left(n+1\right)n{c}_{n+1}{p}_{0}+n{c}_{n}{q}_{0}+{c}_{n-1}{r}_{0}\\ =& \left(n+1\right)n{p}_{0}{c}_{n+1}+\left[n\left(n-1\right){p}_{1}+n{q}_{0}\right]{c}_{n}+\cdots \\ +\left(2{p}_{n-1}+2{q}_{n-2}+{r}_{n-3}\right){c}_{2}+\left({q}_{n-1}+{r}_{n-2}\right){c}_{1}+{r}_{n-1}{c}_{0},\end{array}$
which implies
$\begin{array}{rcl}{c}_{n+1}& =& \frac{1}{\left(n+1\right)n{p}_{0}}{a}_{n-1}-\frac{n\left(n-1\right){p}_{1}+n{q}_{0}}{\left(n+1\right)n{p}_{0}}{c}_{n}-\cdots \\ -\frac{2{p}_{n-1}+2{q}_{n-2}+{r}_{n-3}}{\left(n+1\right)n{p}_{0}}{c}_{2}-\frac{{q}_{n-1}+{r}_{n-2}}{\left(n+1\right)n{p}_{0}}{c}_{1}-\frac{{r}_{n-1}}{\left(n+1\right)n{p}_{0}}{c}_{0}\\ =& {\alpha }_{n+1}^{0}{a}_{0}+{\alpha }_{n+1}^{1}{a}_{1}+\cdots +{\alpha }_{n+1}^{n-1}{a}_{n-1}+{\beta }_{n+1}{c}_{0}+{\gamma }_{n+1}{c}_{1},\end{array}$

where ${\alpha }_{n+1}^{0},\dots ,{\alpha }_{n+1}^{n-1}$, ${\beta }_{n+1}$, ${\gamma }_{n+1}$ are complex numbers. That is, ${c}_{n+1}$ is a linear combination of ${a}_{0},{a}_{1},\dots ,{a}_{n-1}$, ${c}_{0}$, ${c}_{1}$, which ends the proof. □

In the following theorem, we investigate a kind of Hyers-Ulam stability of the linear differential equation (1). In other words, we answer the question whether there exists an exact solution near every approximate solution of (1). Since $x=0$ is an ordinary point of (1), we remark that ${p}_{0}\ne 0$.

Theorem 3.2 Let $\left\{{c}_{m}\right\}$ be a sequence of complex numbers satisfying the recurrence relation (3) for all $m\in {\mathbb{N}}_{0}$, where (b) is referred for the value of ${a}_{m}$, and let ${\rho }_{2}$ be the radius of convergence of the power series ${\sum }_{m=0}^{\mathrm{\infty }}{c}_{m}{x}^{m}$. Define ${\rho }_{3}=min\left\{{\rho }_{0},{\rho }_{1},{\rho }_{2}\right\}$, where $\left(-{\rho }_{0},{\rho }_{0}\right)$ is the domain of the general solution to (1). Assume that $y:\left(-{\rho }_{1},{\rho }_{1}\right)\to \mathbb{C}$ is an arbitrary function belonging to $\mathcal{C}$ and satisfying the differential inequality
$|p\left(x\right){y}^{″}\left(x\right)+q\left(x\right){y}^{\prime }\left(x\right)+r\left(x\right)y\left(x\right)|\le \epsilon$
(7)
for all $x\in \left(-{\rho }_{3},{\rho }_{3}\right)$ and for some $\epsilon >0$. Let ${\alpha }_{n}^{0},{\alpha }_{n}^{1},\dots ,{\alpha }_{n}^{n-2}$, ${\beta }_{n}$, ${\gamma }_{n}$ be the complex numbers satisfying
${c}_{n}={\alpha }_{n}^{0}{a}_{0}+{\alpha }_{n}^{1}{a}_{1}+\cdots +{\alpha }_{n}^{n-2}{a}_{n-2}+{\beta }_{n}{c}_{0}+{\gamma }_{n}{c}_{1}$
(8)
for any integer $n\ge 2$. If there exists a constant $C>0$ such that
$|{\alpha }_{n}^{0}{a}_{0}+{\alpha }_{n}^{1}{a}_{1}+\cdots +{\alpha }_{n}^{n-2}{a}_{n-2}|\le C|{a}_{n}|$
(9)
for all integers $n\ge 2$, then there exists a solution ${y}_{h}:\left(-{\rho }_{3},{\rho }_{3}\right)\to \mathbb{C}$ of the linear homogeneous differential equation (1) such that
$|y\left(x\right)-{y}_{h}\left(x\right)|\le CK\epsilon$

for all $x\in \left(-{\rho }_{3},{\rho }_{3}\right)$, where K is the constant determined in (b).

Proof By the same argument presented in the proof of Theorem 2.1 with ${\sum }_{m=0}^{\mathrm{\infty }}{b}_{m}{x}^{m}$ instead of ${\sum }_{m=0}^{\mathrm{\infty }}{c}_{m}{x}^{m}$, we have
$p\left(x\right){y}^{″}\left(x\right)+q\left(x\right){y}^{\prime }\left(x\right)+r\left(x\right)y\left(x\right)=\sum _{m=0}^{\mathrm{\infty }}{a}_{m}{x}^{m}$
(10)
for all $x\in \left(-{\rho }_{3},{\rho }_{3}\right)$. In view of (b), there exists a constant $K\ge 0$ such that
$\sum _{m=0}^{\mathrm{\infty }}|{a}_{m}{x}^{m}|\le K|\sum _{m=0}^{\mathrm{\infty }}{a}_{m}{x}^{m}|$
(11)

for all $x\in \left(-{\rho }_{1},{\rho }_{1}\right)$.

Moreover, by using (7), (10), and (11), we get
$\sum _{m=0}^{\mathrm{\infty }}|{a}_{m}{x}^{m}|\le K|\sum _{m=0}^{\mathrm{\infty }}{a}_{m}{x}^{m}|\le K\epsilon$

for any $x\in \left(-{\rho }_{3},{\rho }_{3}\right)$. (That is, the radius of convergence of power series ${\sum }_{m=0}^{\mathrm{\infty }}{a}_{m}{x}^{m}$ is at least ${\rho }_{3}$.)

According to Theorem 2.1 and (10), $y\left(x\right)$ can be written as
$y\left(x\right)={y}_{h}\left(x\right)+\sum _{n=0}^{\mathrm{\infty }}{c}_{n}{x}^{n}$
(12)

for all $x\in \left(-{\rho }_{3},{\rho }_{3}\right)$, where ${y}_{h}\left(x\right)$ is a solution of the homogeneous differential equation (1). In view of Lemma 3.1, the ${c}_{n}$ can be expressed by a linear combination of the form (8) for each integer $n\ge 2$.

Since ${\sum }_{n=0}^{\mathrm{\infty }}{c}_{n}{x}^{n}$ is a particular solution of (2), if we set ${c}_{0}={c}_{1}=0$, then it follows from (8), (9), and (12) that
$|y\left(x\right)-{y}_{h}\left(x\right)|\le \sum _{n=0}^{\mathrm{\infty }}|{c}_{n}{x}^{n}|\le CK\epsilon$

for all $x\in \left(-{\rho }_{3},{\rho }_{3}\right)$. □

## Declarations

### Acknowledgements

Dedicated to Professor Hari M Srivastava.

This research was completed with the support of The Scientific and Technological Research Council of Turkey while the first author was a visiting scholar at Istanbul Commerce University, Istanbul, Turkey.

## Authors’ Affiliations

(1)
Mathematics Section, College of Science and Technology, Hongik University
(2)
Department of Mathematics, Faculty of Sciences and Arts, Istanbul Commerce University

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